(i.) It is not necessary to use tongue as well as brain. Do not, therefore, say nine and eight are seventeen, and five are twenty-two, and four are twenty-six, and seven are thirty-three, but go through the same process mentally, as follows:Nine, seventeen, twenty-two, twenty-six, thirtythree. (ii.) If any number occurs more than once in a column make use of multiplication. Thus, in the second column, at a glance there are three eights and two other numbers, making together a fourth eight. The sum of the column is therefore 32, making 35 with the three carried from the first column. (iii.) Group sets of numbers whose sum is ten; these are easily recognised and the addition of ten to a number is simplicity itself. Thus, in the third column, carrying three the process would run as follows:-3, 13 (8 + 2), 23 (7+3). = When adding two numbers one of which is a little less than a power of ten, the process is simplified as follows:-e.g., 893 + 997 890 + 1000 1890, i.e., replace the number in question by the power of ten, and make a balancing reduction in the other number. = In the subtraction of decimals of many places from whole numbers it is advisable to remember that the process of carrying one and then subtracting from ten is equivalent to a subtraction from 9. Thus subtract 8.4379246 from 9 3 First, to obtain the units, multiply units by units. 4X7: 28. Put down the 8 and carry the 2 to the tens stage. The other tens are obtained by multiplying the units in one number by the tens in the other. Thus 2+(4x8)+(5×7)=69. Put down the 9 and carry the 6 to the hundreds which are obtained by multiplying the tens digits together. 6+(5×8)=46. After a little practice the whole process can be done very rapidly. Example: multiply 256 by 394. : It is well to remember this when dealing with the logarithms of numbers less than unity where, though the logarithm is negative the mantissa is to be positive, the characteristic being a negative whole number. Thus 4879632 = 15120368 and vice versa. Similarly negative whole numbers may be transformed into positive with a compensating negative power of ten: e.g., -78963 121037. This is sometimes useful when finding the sum of a number of positive and negative numbers. All the negative numbers are transformed by the above method, and the sum is reduced to an addition. 8 6 4 0 8 6 4 100864 [7+8+45 [7+18+15 [4+6=10] +18=78] =40] Here the process is similar. It is well to bear in mind the diagrams formed by the arrows. They are symmetrical, and are easily remembered. The units and tens are obtained in the same way as in the first example, but hundreds can be obtained both by multiplying tens digits together and by multiplying the hundreds digits in one number by the units digits in the other. Thousands are obtained by the product of the tens digits with the hundreds digits, and tens of thousands by the product of the two hundreds digits. The process can of course be continued to higher numbers, but it becomes rather complicated in such cases. Certain cases of multiplication by numbers of special form call for notice. (1) To multiply by a power of ten add as many noughts to the multiplicand as there are noughts in the multiplier. (2) Since 510, 25 100 and 1909, multiply by 5, 25 and 125 add one, two or three noughts, as the case may be, and divide by 2, 4 or 8. (3) If the multiplier is a little less than a power of ten, e.g., 995 = 1000 5, proceed as follows:8976 × 995=8976 × (1000−5) = 8976000 - (89760) = 8976000 49880 8926120 A MODERN-LANGUAGE MASTER'S LIBRARY OF FRENCH AND GERMAN. By OTTO SIEPMANN. Head of the Modern-Language Department at Clifton College. I' German. Na previous article which appeared in this Magazine (March, 1901), I have laid stress on the necessity that a young modern-language master who wishes to equip himself for his difficult task must, in the first place, try to acquire a good pronunciation. It is sometimes thought that it is much easier for an Englishman to learn to pronounce German correctly than French; and, in consequence, students and teachers do not take the same amount of trouble in trying to master the sounds of German, and are satisfied with a lower standard of correctness. This is a grave mistake. Even if it were true that German sounds come more naturally to the English organs of speech, there can be no doubt that the difficulties of a really good German pronunciation are so great as to make the study of phonetics indispensable. It is a delusion and a snare to believe that a short stay in Germany will be sufficient to set things right in matters of pronunciation. My advice to young modern-language masters is to study carefully one of the books on general phonetics, and of French and German in particular, which I have mentioned in my last article, and add to this Vietor's excellent little treatise, entitled "GERMAN PRONUNCIATION, PRACTICE, AND THEORY," 2nd edition (Leipzig, 1899, 2s.), and Georg Hempl's "German Orthography and Phonology" (Strassburg, Trübner). After these, Bremer's "Deutsche Phonetik (Leipzig, 1893, 5s.) will be found interesting and stimulating. It goes without saying that a few hours' help from a native with a keen ear for sound differences and a good knowledge of phonetics will be found invaluable at the initial stages; but it may be pointed out that a German with an untrained ear and no acquaintance of phonetics will in all probability be found of little use; in fact, it is not impossible that his observations may prove positively misleading, for as a rule the statements of such natives about their own pronunciation and manner of producing speechsounds are most untrustworthy, and when they proceed to correct the pronunciation of a foreigner their criticism and advice are as a rule of a most deplorable kind. If a thoroughly competent native cannot be found, the student should try to obtain the help of an Englishman who knows German and phonetics. Along with the study of phonetics and its practical application, which may quite well be carried on in one's study, the mastery of grammar and wide and deep reading are considerable aids for future proficiency as a teacher. Assuming that one of the ordinary schoolgrammars, such as those by Eve, Aue, Meissner, Weisse, Kuno Meyer or Whitney, is in the teacher's possession, he might with advantage take in hand. "A GRAMMAR OF THE GERMAN LANGUAGE," by H. C. G. Brandt (Boston, Allyn & Bacon, about 4s. 6d.), and study along with it Heyse's" Deutsche Grammatik oder Lehrbuch der deutschen Sprache," 25th edition, by Dr. Otto Lyon (Leipzig, 1893, about 4s.), or "Handbuch der deutschen Sprache,' von Karl F. Becker, 11th ed. (Prag, 1876, about 5s.), which should be supplemented by a little work entitled "SATZBAU UND WORTFOLGE IN Der DEUTSCHEN SPRACHE," VON PROF. DR. DANIel SANDERS (Berlin, 1883, about 2s. 6d.). A more scientific work on the same subject is "Der Hermann deutsche Satzbau," dargestellt von Wunderlich (Stuttgart, 1892, 252 pages), of which a very much enlarged second edition is appearing in two vols.; vol. i., 9s. unbound. The most complete work on modern German grammar I know is the "Neuhochdeutsche Grammatik mit Berücksichtigung der historischen Entwickelung der deutschen Sprache," von Friedrich Blatz, 3rd edition (Karlsruhe, 1895, vol. i., 856 pp., 10s.; vol. ii., 1,314 pp., 15s.). With these works, a taste for the historical treatment of the German language is sure to develop, and BEHAGHEL, DIE DEUTSCHE SPRACHE (Leipzig, 1887, s.) may be recommended as a guide for beginners. Of this there exists a good English translation by Trechmann (London, Macmillan). Wasserzieher's booklet "Aus dem Leben der deutschen Sprache" (Leipzig, 64 pp., stitched, for 4d.) is very small, but the student who buys it certainly gets his money's worth. Before mentioning any of the larger historical works, I must not omit two books which are excellent food for those who like to leave the beaten track and to think for themselves; they are Kern's "Deutsche Satzlehre" (Berlin, 1888, 184 pp.) and Ries' "Was ist Syntax"? (Marburg, 1894, 163 pp.). A study of these cannot fail to arouse the teacher's argumentative faculties, and he will be quite prepared to enjoy some other books of a critical description written by men who are dissatisfied with much that is used in present-day German, and aim at reform. "Allerhand Sprachdummheiten-Kleine deutsche Grammatik des Zweifelhaften, des Falschen und Hässlichen—Ein Hilfsbuch für alle, die sich öffentlich der deutschen Sprache bedienen" von Dr. Gustav Wustmann (Leipzig, 1891, 320 pp., 2s.)-is a worthy lead off in this group of interesting publications. Erbe's Rand bemerkungen" (Stuttgart, 1892, 50 pp., 6d.), Minor's "Allerhand Sprachgrobheiten (10d.) and a host of other "Streitschriften followed in its train. Another book, Sprachleben und Sprachschäden," by Matthias, though not directly connected with the polemic raging round Wustmann's excellent little work, should be read by every teacher of German, and in conjunction with it Lehmann's "Sprachliche Sünden der Gegenwart" (Braunschweig, 1882, 175 pp., about 2s.), Heintze's "Gut Deutsch" (Berlin, 1898, 200 pp., about 2s.), Brunner's "Schlecht Deutsch" (Leipzig, 1895, 199 pp., about 25.), and Schroeder's "Vom papiernen Stil " (Berlin, 1896, 102 pp., about 2s.). "Sprachgebrauch und Sprachrichtigkeit" von Andresen (8th ed., Leipzig, 1899, 430 pp., 5s.) is a standard work which needs no recommendation. A stimulating volume is Reichel's "Sprachpsychologische Studien." Vier Abhandlungen über Wortstellung und Betonung des Deutschen in der Gegenwart, Sparsamkeit, Begründung der Normalsprache (Halle, 1897, 337 pp., about 4s.). And most excellent is Sütterlin's new work "Die deutsche Sprache der Gegenwart (1900). There are some works on composition and style of Modern German which are most suitable for teachers, of which I must mention a few:— "DER DEUTSCHE AUFSATZ," VON ERNST LAAS, 2nd ed. of Part I. (Berlin, 1877); 3rd ed. of Part II., by Imelmann (Berlin, 1894, price about 10s.); "Der deutsche Stil von R. F. Becker, neubearbeiter," von Dr. Otto Lyon, 3rd ed. (Leizig, 1884, about 6s.); Sanders' "Deutsches Stil Musterbuch (Berlin, 1886, about 5s.); "DIE SPRICHWÖRTLICHEN REDENSARTEN VON BORCHARDT," in gänzlicher Umarbeitung, herausgegeben von Gustav Wustmann, 5th ed. (Leipzig, 1895, about 6s.); and "Der Bilderschmuck der deutschen Sprache in tausenden volkstümlicher Redensarten," nach Ursprung und Bedeutung erklärt von Dr. Hermann Schröder, 5th ed. (Weimar, 1896, about 6s.). I would also advise every master or mistress of German to become a member of "Der Allgemeine Deutsche Sprachverein," which publishes a Zeitschrift, with valuable articles for teachers, and also "Wissenschaftliche Beihefte," with excellent contributions by eminent authors. With "Unsere Muttersprache, ihr Werden und ihr Wesen," von Prof. Dr. O. Weise (Leipzig, 2nd ed., 1896, 270 pages, about 3s. 6d.) we enter again upon the historical platform, which may induce teachers to embark on the study of some of the greatest works, such as Von der Gabellentz' "Die Sprachwissenschaft" (14s.), Wundt's "Völkerpsychologie, I. Sprache (14s.) [which should be supplemented by Delbrück's "Grundfragen der Sprachforschung" (1901, 45.)], or at any rate, of Whitney's "Life and Growth of Language" (6s.), and the interest taken in these philosophical works will most probably whet the appetite for a more thorough study of historical grammar, for which Cerf's "Short Historical Grammar of the German Language "(Williams & Norgate, 1894, Part I., Introduction and Phonology), and PROF. JOSEPH WRIGHT'S "GOTHIC, OLD AND MIDDLE HIGH GERMAN PRIMERS" will be found excellent helps and, at the same time, a good introduction to the study of the "Deutsche Grammatik (Gotish, Alt-, Mittel-, und Neuhochdeutsche "), von W. Wilmann's Erste Abteilung-Lautlehre (Strasburg, 1893, 332 pages, about 6s.), Zweite AbteilungWortbildung (Strassburg, 1896, 663 pages, about IOS.), as well as to Erdmann-Mensing's Grundzüge der deutschen Syntax." The most excellent books on individual dialects for ad vanced students are found in Braune's " Sammlung kurzer Grammatiken Germanischer Dialekte" (Halle, Max Niemeyer). Braune's "Gotische Grammatik" (Halle, 1882, 2 aufl., 129 pp., about 3s. 6d.) might be preceded by Sheitberg's "Gotisches Elementarbuch (3s. 6d.), Braune's "Althochdeutsche Grammatik" (Halle, 2nd. ed., 5s. 6d.), Paul's "Mittelhochdeutsche Grammatik" (Halle, 1884, about 3s. 6d.). Braune's "Althoch 1 The annual subscription is 5s. Intending members should write to the President of the London Branch, Professor Weiss, Royal Military Academy, Woolwich, or the Secretary, Dr. Hirsch, 102, Embleton Road, Lewisham, London, S.E. deutsches Lesebuch (Halle, 1881, 2nd. ed., 3s. 6d.) are the most reliable works on the older dialects. 66 Bachmann, "Mittelhochdeutsches Lesebuch mit Grammatik und Wörterbuch" (Zürich, 1892, about 5s.) is a suitable book for teachers who have not sufficient time to study the older portions of the German language and literature fully. There are several other first-rate books which form good introductions; such are Heiderich, “ Einführung in das Studium des Gotischen" (Is. 3d.), for beginners, then “Zupitza,” “ Einführung in das Studium des Mittelhochdeutschen zum Selbstun. terricht für jeden Gebildeten," 5th ed., by Nobiling (Oppeln, about 3s. 6d.), also Michels, "Mittelhochdeutsches Elementarbuch," in "Sammlung von Elementarbüchern der altgermanischen Dialekte Herausgeben von Dr. W. Streitberg" (Heidelberg, 5s.). For the transition from Middle High German to Modern High German the following will be of good service: Arndt, "Der Übergang vom Mittelhochdeutschen zum Neuhochdeutschen der Breslauer Kanzlei " (5s.), Rückert, "Geschichte der neubochdeutschen Schriftsprache," 2 vols., 7s. each, and von Bahder, "Grundlagen des neuhochdeutschen Lautsystems." Teachers who have no wish to study the older dialects I would strongly urge to take up at least the following 10d. editions of the "Sammlung Göschen" :—“Gotische Sprachdenkmäler übersetzt und erläutert; Althochdeutsche Literatur mit Grammatik, Übersetzung und Erläuterung";-Hartmann von Aue, Wolfram von Eschenbach und Gotfried von Strassburg;— Waltharilied Walther von der Vogelweide, Minnesang und Spruchdichtung - Der Nibelunge Nôt Kudrun und Dietrichepen-and Sebastian Brant, Hans Sachs, Luther, Fischart. Others may wish to go further afield to Paul's "Altdeutsche Textbibliothek”: (1) "Die Gedichte Walthers von der Vogelweide," hrg. von Paul, 1882, 1s. 10d. (2) "Gregorius von Hartmann von Aue, hrg. von Paul, 1882, Is. (3) "Der arme Heinrich" von Hartmann, von Aue, von Paul, 1882, 6d. (4) “Heliand," hrg. von Behaghel, 1882, 2s. 6d. (5) "Kudrun," hrg. von B. Symons, 1883, 2s. 10d., and others, or to editions of the great masterpieces of the older German literature. I would mention "Wolfram von Eschenbach, Fünfte Ausgabe," von Karl Lachmann, Berlin, 1891, 640 pp.; Iwein, "Eine Erzählung von Hartmann, von Aue, mit Anmerkungen," von G. F. Benecke und K. Lachmann, zweite Ausgabe (Berlin, 1843, 565 pp.); "Der Nibelunge Nôt, mit den Abweichungen von der Nibelunge Liet, den Lesarten sämtlicher Handschriften und einem Wörterbuch," herausgegeben von Karl Bartsch (I. teil text, Leipzig, 1870, 394 pp. ; II. teil, Erste Hälfte, Lesarten, Leipzig, 1876, 292 pp.; II. teil, Zweite Hälfte Wörterbuch, Leipzig, 1880, 411 pp.); “Kudrun," herausgegeben und erklärt, von Ernst Martin (Halle, 1872, 336 pp.); "Walther von der Vogelweide, by Wilmanns (2nd. ed., Halle, 1883) in Zacher's "Germanistische Handbibliothek." Wilmanns has also published "Leben und Dichtungen Walther's," von der Vogelweide (Bonn, 1882). For the sixteenth and seventeenth centuries I would recommend the "Liederbuch aus dem sechzehnten Jahrhundert von Goedeke und Tittmann (2nd. ed., Leipzig, 1881, 400 pp., 3s. 6d.), and the same authors' "Deutsche Dichter des sechzehuten und siebzehnten Jahrhunderts mit Einleitung und Anmerkungen" (each volume 3s. 6d.). Then for the eighteenth century Seuffert's excellent collection of "Deutsche Literaturdenkmale des 18 Jahrhunderts." The editions of the great classical writers, Klopstock, Wieland, Lessing, Herder, Goethe, Schiller, Grillparzer, Heine, are so numerous that it is difficult to make a selection, and most teachers will not be able to purchase and work through the complete works of even the greatest writers. How lin, Hertz, 1896), C. F. MEYER'S "JÜRG JENATSCH und Romanzen (Macmillan); and read a few modern plays, such as WILDENBRUCH'S "HAROLD" and "König Heinrich," SUDERMANN'S "Johannes,' "EHRE" or "HEIMAT," HAUPTMANN'S WEBER or "Einsame Menschen." These will give the reader an idea of the tendencies and struggles of modern German literature. ever, they should use their influence that the best editions of the most eminent representatives of German literature are procured for their school libraries, such as the scholarly works of Old and. Middle High German mentioned above, and then some, at least, of the following: Luther's Werke, "Kritische Gesamtausgaube' (Weimar), Lessing's "Sämtliche Schriften herausgegeben von Karl Lachmann," 3rd edition, by Franz Muncker (Stuttgart, since 1886); Herder's "Sämtliche Werke herausgegeben,' von Bernhard." Deutsche Lyrik" and Buchheim's "Balladen Suphan; "Goethe's Werke herausgegeben im Auftrage der Grossherzogin Sophie von Sachsen," the so-called Weimar edition; Schiller's "Sämtliche Schriften herausgegeben von Karl Goedeke"; Heine's Werke von Dr. E. Elster,; Grillparzer's Sämtliche Werke, 20 vols., herausgegeben und mit Einleitungen versehen von August Sauer. The latter edition forms part of the "COTTA'SCHE Bibliothek DER WELTLITERATUR," in which most of the other classics have also appeared. They are very convenient editions for teachers, and can be bought separately, each volume bound in cloth, for Is. Or cheaper still are the paper-cover editions of "RECLAM'S UNIVERSALBIBLIOTHEK," which cost 3d. each. Along with any of these should be used Düntzer's "Erläuterungen zu den deutschen Klassikern" (Leipzig, about is. each). Students of German who find it more convenient to use editions with English notes cannot do better than take up Breul's editions of some of the German classics for the Pitt Press, and Buchheim's editions for the Clarendon Press. I may also mention Rippmann's edition of Grillparzer's Sappho," and Cotteril's edition of 'Iphigenie" in my own series; and there are many excellent edition's in Macmillan's German Classics, edited by Fassnacht, among them GOETHE'S "FAUST," Part I. Teachers who wish to study this great work thoroughly should take the edition by Colvin Thomas (Boston, Heath & Co.), which is most scholarly, and covers both the first and second part, as does Prof. Schröer's edition of "Faust," Part I., 5s.; Part II., 6s. 6d. 66 Some of the editions mentioned contain biographies of the great writers, but they are necessarily short and sketchy. Those who wish to study the lives of the great German classics more deeply I would refer to Erich Schmidt's "Lessing, Geschichte seines Lebens und seiner Schrifteu" (Berlin, 1884, 2 vols.), Bielschowsky's "Goethe," 2 vols., or Eugen Wolff's "Goethe's Leben und Werke," or R. M. Meyer's preisgekrönte Arbeit "Goethe" (628 pages), also Düntzer's "Goethe's Leben," 10s. should be mentioned, and perhaps Lewes' "Life of Goethe." Among the biographies of Schiller, Minor's "Schiller, sein Leben und seine Werke" (Berlin, Weidmann) takes the first place; Weltrich, Brahm and Düntzer have likewise furnished valuable contributions. If I am to give some guidance to the more recent works of German literature I cannot do better, I think, than quote a few authors and works without wishing to say that there are not many others equally good. Take GUSTAV FREYTAG'S "ÁHNEN," SCHEFFEL'S" EKKEHARD " (Stuttgart, 1893), GOTTfried Keller'S "ZÜRICHER NOVELLEN (Ber Turning to the history of the German language and literature, I would call special attention to the "Histoire de la langue allemande" par Henri Lichtenberger (Paris, 1895, 478 pages), which is likely to suit many teachers who prefer to read a book written in French. Among histories of literature I have so many good works before me on my shelves that the embarras de richesse which has been troubling me all along is taking a very acute form, and I hope readers of this article will forgive me if I call their attention to some works which they may very likely not think of purchasing. Let me begin with a book written in English, entitled "Outlines of German Literature," by Gostwyck and Harrison, of which, I think, a new revised edition by a competent German scholar will appear before very long (London, 1873, 10s.). Of Hermann Kluge's "Geschichte der deutschen National-Literatur" (Altenburg, 22nd ed., 1891, about 2s. 6d.), there is, I believe, also an English translation. Excellent little books are 'Grundzüge der deutschen Litteraturgeschichte" von Dr. Gotthold Klee (Berlin, 3rd ed., 1898, about 3s.), and “Grundüge der deutschen Litteraturgeschichte von Dr. Gottlob Egelhaaf" (Heilbronn, 6th ed., 1888, about 2s. 6d.). Larger works and more suitable for serious students are:-Wilhelm Scherer's "Geschichte der deutschen Litteratur " (Berlin, 8th ed., 12s.), of which there exists an English translation by Mrs. Conybeare. Koberstein's "Grundrisz der Geschichte der deutschen Nationalliteratur," in 5 vols. (5th ed., 1892, by Karl Bartsch), is a most thorough work. An edition with many facsimiles and illustrations is König's "Deutsche Litteraturgeschichte," which costs about £I. A similar and in its text more scholarly work is the "Geschichte der deutschen Litteratur von den ältesten Zeiten bis zur Gegenwart" by Vogt and Koch (Leipzig, 1867, 16s.). A standard work for advanced students who wish to study the subject thoroughly is Karl Goedeke's "Grundrisz zur Geschichte der deutschen Dichtung" (2nd edition, after the death of the author continued by E. Goetze). All that a teacher may require by way of a history of the German literature of the older period is contained in "Paul's Grundrisz der Germanischen Philologie," of which a second edition is appearing, and without which no serious student of German can exist. For the eighteenth and nineteenth centuries a few special works will be necessary. For the eighteenth century Hermann Hettner's "Geschichte der deutschen Litteratur im XVIIIten Jahrhundert " (2 vols., 4th edition, by Harnack) can be thoroughly recommended. There is as yet no standard book dealing with the nineteenth century, but Rudolf von Gottschall "Die Deutsche Nationalliteratur des neunzehnten Jahrhunderts "(6th ed., 4 vols, 1892), and |